Principles and Practices of Molecular Properties: Theory, Modeling, and Simulations

By Patrick Norman, Kenneth Ruud and Trond Saue

A comprehensive yet accessible exploration of quantum chemical methods for the determination of molecular properties of spectroscopic relevance

Molecular properties can be probed both through experiment and simulation. This book bridges these two worlds, connecting the experimentalist's macroscopic view of responses of the electromagnetic field to the theoretician’s microscopic description of the molecular responses. Comprehensive in scope, it also offers conceptual illustrations of molecular response theory by means of time-dependent simulations of simple systems.

This important resource in physical chemistry offers: 

• A journey in electrodynamics from the molecular microscopic perspective to the conventional macroscopic viewpoint
• The construction of Hamiltonians that are appropriate for the quantum mechanical description of molecular properties
• Time- and frequency-domain perspectives of light–matter interactions and molecular responses of both electrons and nuclei
• An introduction to approximate state response theory that serves as an everyday tool for computational chemists
• A unified presentation of prominent molecular properties

Principles and Practices of Molecular Properties: Theory, Modeling and Simulations is written by noted experts in the field. It is a guide for graduate students, postdoctoral researchers and professionals in academia and industry alike, providing a set of keys to the research literature.

 

• Language: English
• Publisher: Wiley
• Release date: Jan 15, 2018
• ISBN: 9781118794814

Book preview

Chapter 1

Introduction

If you want to find the secrets of the universe, think in terms of energy, frequency and vibration.

Nicolas Tesla

This is a book about molecular properties, or to be more specific, molecular response properties. Response properties tell us about how molecules respond to electromagnetic fields. To understand these responses, we have to enter the microscopic world of atoms and molecules, governed by the laws of quantum mechanics. For that reason, the reader of this book can expect several intellectual challenges ranging from profound and conceptual cornerstones of quantum theory itself to trivial, yet mind-boggling, issues relating to the smallness of atomic sizes. Consider for instance the situation in which a collection of molecules are being exposed to the intense electric field of a laser, as illustrated in Figure 1.1. From a human perspective, the focal point of a laser is a dangerous place to be, but, from the atomic perspective, it is far less dramatic. In our example, there will be fewer photons than molecules, and, for instance, if the purpose is to protect the eye by efficient optical power limiting, only about every second molecule needs to absorb an incoming light quanta in order to reduce the energy in the transmitted light pulse to an eye-safe level. Furthermore, as strong as the electric field may appear to our eyes, to the individual electron it is several orders of magnitude smaller than the dominating forces exerted by the atomic nuclei and fellow electrons. To get an idea of magnitudes, one may note that the electric field below overhead power lines may reach c01-math-001 and the maximum electric field strength possible in air without creating sparks is c01-math-002 . In contrast, at the Bohr radius c01-math-003 in the hydrogen atom, the electric field strength is c01-math-004 . This is a key point, namely, that we can expose molecules to fields that are strong enough so that we can detect the responses of their charges (nuclei and electrons) while at the same time the fields are weak enough to act as probes, not significantly perturbing the electronic and nuclear structure of the molecule.

Geometrical illustration of Liquid benzene in a small volume corresponding to the focal point of a laser operating at 532 nm and releasing pulses with an energy of 1 mJ.

Figure 1.1 Liquid benzene in a small volume corresponding to the focal point of a laser operating at 532 nm and releasing pulses with an energy of 1 mJ.

Take a very simple example: What happens if a neutral atom (not even a molecule) is placed in a uniform electric field? An experimentalist will ask nature—that is, he or she may perform an experiment, where every macroscopic experiment relates to a very large number of probabilistic microscopic quantum events—by probing how the charge distribution of the atom is modified by the applied field. A theoretician will ask the wave function c01-math-005 . The quantum-mechanical equivalent to the outcome of the experiment is the expectation value

1.1 equation

where c01-math-007 is the quantum-mechanical operator corresponding to the observable monitored by the experiment. Quantum mechanics is a probabilistic theory. The link between theory and experiment is made by considering a large number of systems prepared in the same state, prior to switching on the field. If we disregard measurement errors, then the possible outcomes of the individual quantum events are given by the eigenvalues c01-math-008 of the operator c01-math-009 , defined by the eigenvalue equation

1.2 equation

Following the postulates of quantum mechanics, the operator c01-math-011 is by necessity Hermitian, and the eigenvalues are thus real (corresponding to real-valued observables), and there is a probability c01-math-012 for the outcome c01-math-013 in each of the single quantum events, leading to an expectation value that is

1.3 equation

For example, indirect information about the charge distribution of the atom can be obtained from measurements of the electric dipole moment since the two quantities are connected through an expectation value of the form

1.4

equation

where c01-math-016 denotes the number of electrons and c01-math-017 is the elementary charge. However, the electronic charge density can in itself also be expressed as an expectation value

1.5 equation

and it is possible to probe c01-math-019 in for instance X-ray diffraction experiments.

Geometrical illustration of Electronic charge density of neon expanded in orders of the applied electric field F.

Figure 1.2 Electronic charge density of neon expanded in orders of the applied electric field c01-math-020 . Light and dark gray regions indicate positive and negative values, respectively.

If the external electric field is weak compared to the internal atomic fields, we can expand the induced electronic charge density in a Taylor series with respect to field strength. In Figure 1.2, such a perturbation expansion is illustrated to fifth order for a neon atom. The electric field of strength c01-math-021 is applied along the vertical c01-math-022 -axis (directed upward in the figure) and will tend to pull the positive charge along the field and the negative charge in the opposite direction, resulting in an electronic charge density that can be expanded as

1.6

equation

The zeroth-order density c01-math-024 refers to that of neon in isolation and integrates to c01-math-025 . It follows from charge conservation that the higher-order densities all integrate to zero. The first-order density c01-math-026 shows the charge separation of a dipole, and we then get more and more complicated structures with increasing order. It is also clear that the higher the order, the more diffuse the density becomes, and we can expect that an accurate description of higher-order responses put strong requirements on the wave function flexibility at large distances from the nucleus.

If we insert the expansion of the charge density into the expression for the dipole moment [Eq. (1.4)], even orders of the density will not contribute due to symmetry—this is a reflection of the fact that odd-order electric properties vanish in systems with a center of inversion. The resulting induced dipole moment, directed along the c01-math-027 -axis, becomes

1.7

equation

This expression defines a series of proportionality constants between the induced dipole moment and powers of the field. The linear and cubic coupling constants are known as the electric dipole polarizability and second-order hyperpolarizability, and they are conventionally denoted by Greek letters c01-math-029 and c01-math-030 , respectively.¹ It is the focus of this book to understand how these and other molecular properties can be determined by means of quantum-chemical calculations.

Geometrical illustration of Hierarchy of quantum-chemical methods.

Figure 1.3 Hierarchy of quantum-chemical methods.

When judging the quality of quantum-chemical calculations, one typically considers the choice of method and basis set. These two quantities combined constitute a theoretical model chemistry, that is, a certain approximation level reaching toward the exact solution of the electronic wave function equation. There exist hierarchical series of basis sets that allow for systematic convergence toward the complete one-particle basis set limit, as indicated in Figure 1.3. An increase in the cardinal number of the basis set, from double- to triple- c01-math-031 and so forth, improves the description of the ground-state wave function, whereas levels of augmentation with diffuse functions in the basis set are particularly important for the description of the excited electronic states, and therefore also for many molecular properties. Likewise, in conventional wave function-based electronic structure theory, the configuration interaction (CI) and coupled cluster (CC) expansions provide systematic ways to reach the complete c01-math-032 -particle limit. Increased complexity of the theoretical model chemistry comes, however, at a sometimes staggering computational cost. In general, the computational cost scales as c01-math-033 , where the base c01-math-034 represents the size of the one-particle basis set, and therefore implicitly scales with the system size, and the exponent c01-math-035 is associated with a given electronic structure method. Starting from the Hartree–Fock (HF) method, which formally scales as c01-math-036 , each excitation level treated variationally (perturbatively) increases the exponent by two units (one unit). Accordingly, CC and CI models that include single and double (SD) excitations, CCSD and CISD, respectively, scale as c01-math-037 , but the CC expansion includes electron correlation in a more efficient manner than does CI and has other advantages such as size extensivity. Adding triple excitations perturbatively, as in CCSD(T), increases the exponent to seven. A great achievement of quantum chemistry has been to devise algorithms that significantly reduce these formal scalings.

Kohn–Sham density functional theory (KS-DFT) has become the most widely used method in quantum chemistry due to its efficient treatment of electron correlation at modest computational cost. It formally has the same scaling as HF theory as it also employs a single Slater determinant to describe the reference state of the fictitious noninteracting KS system, constrained to have the same electron density as the real interacting system. The similarity in the parametrization of the reference state has implications in the presentation of the time-dependent response approaches. Until the very final stages, we need not specify which of these two approaches we address, treating them instead in a uniform manner. We will present this formulation, common to time-dependent HF and KS-DFT theories, under the name self-consistent field (SCF) theory. At the end of the day, there is only one drawback that stands out as critical with the KS-DFT technique, and that is the lack of a systematic way to improve the exchange-correlation (XC) functional, which makes it impossible to provide a general ranking of DFT functionals. This leads to the necessity to benchmark the performance of different XC functionals, not only with respect to classes of molecular systems but also with respect to molecular properties.

Theoretical model chemistries are represented as points in the two-dimensional space spanned by the axis of electronic structure methods and that of the basis sets, as shown in Figure 1.3. Methods and basis sets should be chosen in a balanced manner. For instance, combining the HF method with a quadruple- c01-math-038 basis set is normally a waste of resources and time. On the other hand, in the 1980s, a widely used theoretical model chemistry was the CISD method combined with double- c01-math-039 basis sets. With increased computational power, however, it was shown numerically that such a combination of methods and basis sets provided reasonable results largely due to a fortuitous cancellation of errors—by increasing the quality of basis sets, the theoretical results moved away from the experimental ones, revealing the intrinsic error of the method. In general, triple- c01-math-040 basis sets or better should be used with CC and CI methods to provide an adequate description of the electron–electron cusp region. It should also be noted that standard basis sets are, in most cases, optimized according to energy criteria and that basis set requirements therefore may (and usually do) change when calculating molecular properties. This is easily understood from Figure 1.2 that demonstrates how the induced charge densities governing molecular properties become increasingly more diffuse as compared to the unperturbed density governing the energy.

Graphical illustration of Speed of 1s-electrons relative to the speed of light (b) and the corresponding Lorentz factor (a) versus atomic number.

Figure 1.4 Speed of c01-math-041 -electrons relative to the speed of light (b) and the corresponding Lorentz factor (a) versus atomic number. Relativistic contractions (c), defined as ratio between relativistic and nonrelativistic HF radial expectation values, of valence c01-math-042 -orbitals for elements of rows 4, 5, and 6 of the periodic table. Atomic numbers c01-math-043 are given relative to atomic number c01-math-044 of the coinage metal of each row (M=Cu, Ag, Au).

Since the 1980s, it has become increasingly clear that a theoretical model chemistry providing a balanced description of all elements of the periodic table requires the inclusion of relativistic effects. Relativistic effects are normally associated with particles moving at speeds close to the speed of light c01-math-045 , which is the case for electrons in the vicinity of heavy nuclei. The kinetic energy of a c01-math-046 -electron in a hydrogen-like ion is

1.8 equation

It is thus proportional to the square of the nuclear charge c01-math-048 and ranges from 13.6 eV for hydrogen to about c01-math-049 eV for the heaviest elements with c01-math-050 . In Eq. (1.8), we have introduced the electron mass c01-math-051 , the Dirac constant c01-math-052 , also known as the reduced Planck constant, and the quantum number c01-math-053 for the electron. For small kinetic energies, when

1.9 equation

we thus expect a linear dependence between the electron speed and the nuclear charge c01-math-055 , as illustrated in Figure 1.4(b). At high kinetic energies, on the other hand, this classical relation breaks down and we have instead

1.10 equation

where the Lorentz factor c01-math-057

1.11 equation

will increasingly deviate from its nonrelativistic limiting value of 1, as illustrated in Figure 1.4(a). In the region of gold (atomic number 79) in the periodic table, the speed of c01-math-059 -electrons reaches values of about c01-math-060 , corresponding to a Lorentz factor of about 1.15. The Lorentz factor appears in the expression for the relativistic mass increase c01-math-061 , and, since the Bohr radius is inversely proportional to the mass, we expect relativity to contract and stabilize orbitals.

One can of course argue, as did Dirac himself, that our focus on the c01-math-062 -orbitals is an extreme case for relativistic effects, and, because the c01-math-063 -electrons orbit very close to their respective nuclei, such relativistic effects will play an insignificant role in most of chemistry. However, relativistic effects propagate out to the valence through their modification of orbitals in the core region and, through the requirement of orthogonality, among all orbitals. An illustration of this aspect is given in Figure 1.4(c), in terms of the contraction of the valence c01-math-064 -orbitals, which is seen to be particularly pronounced for the coinage metal atoms. The direct effect of relativity is to contract orbitals, but this contraction leads to increased screening of the nuclear charge, so there will be a competition between the direct and indirect effects of relativity. In practice, one finds that c01-math-065 - and c01-math-066 -orbitals tend to contract while c01-math-067 - and c01-math-068 -orbitals tend to expand.

Relativistic effects associated with the relativistic mass increase of electrons discussed earlier are denoted scalar relativistic effects. A second relativistic effect is spin–orbit interaction. The name and typical operator form c01-math-069 suggests an interaction due to a coupling of the spin and angular momentum of the electron. However, this confuses the underlying physics that is simply magnetic induction. Spin–orbit coupling refers to the interaction of the electron spin with the magnetic fields generated by charged particles (nuclei or electrons) in relative motion. The orbital angular momentum operator c01-math-070 appears in the operator describing the spin–orbit interaction induced by nuclei because it encodes the relative motion between the electron and the nucleus. Spin–orbit interactions constitute one of the several categories of intramolecular magnetic interactions that are unaccounted for in a nonrelativistic formulation, and they are often introduced in the nonrelativistic framework in an ad hoc manner, as they govern important phenomena in chemistry such as intersystem crossings and phosphorescence. Another example is given by electron paramagnetic resonance (EPR) spectroscopy where one of the parameters, the anisotropy of the molecular c01-math-071 tensor, is rooted in spin–orbit interactions. The spin–orbit interaction couples spin and spatial degrees of freedom, such that spin is no longer a good quantum number. Atomic orbitals will be characterized by their total angular momentum c01-math-072 with possible magnitude c01-math-073 and corresponding azimuthal quantum number c01-math-074 . As an example, the six degenerate c01-math-075 -orbitals split into two degenerate c01-math-076 - and four degenerate c01-math-077 -orbitals. The energy associated with the c01-math-078 -orbitals is lowered as a consequence of an optimum orientation of electron spin in space, see Figure 1.5(a). This flexibility for the direction of the spin to vary between different points in space is not accounted for in the nonrelativistic realm, where orbitals are either pure c01-math-079 - or pure c01-math-080 -spin orbitals. Interestingly, this is also the case for the relativistic c01-math-081 -orbital with c01-math-082 , as illustrated in Figure 1.5(b). The c01-math-083 orbital is pure c01-math-084 simply because c01-math-085 spin ( c01-math-086 ) would require c01-math-087 , which is not a valid quantum number for c01-math-088 angular functions.

Graphical illustration of Hydrogen spin densities in the xz-plane for the (a) 2p1/2 (mj = 1/2) and (b) 2p3/2 (mj = 3/2) orbitals.

Figure 1.5 Hydrogen spin densities in the c01-math-089 -plane for the (a) c01-math-090 and (b) c01-math-091 orbitals.

Dealing with elements from large parts of the periodic table thus corresponds to a situation in which conventional nonrelativistic calculations of the electronic structure may make little sense and relativistic modifications of the Hamiltonian are required, corresponding to the third axis in Figure 1.3. Scalar relativistic effects can be introduced without significant additional computational cost, whereas the inclusion of spin–orbit interactions requires Hamiltonians based on the four-component Dirac operator or two-component approximations to it. A simple way to introduce relativistic effects is to replace the relativistic core orbitals by effective core potentials, but this choice is only viable when is concerned with valence electron properties.

The choice of Hamiltonian is in principle one independent of the two other axes of our chart, which means that the conventional theoretical model chemistries known to us from the nonrelativistic realm can be extended to the relativistic domain. Another important consequence is that the choice of Hamiltonian only comes with a prefactor, independent of system size, to the overall computational cost.

At this point, some readers may feel that a lot of time has been spent discussing relativistic effects, especially if their focus is on molecular systems containing light atoms only. However, it should be kept in mind that a book about molecular properties is essentially about electromagnetic interactions, and electrodynamics—as elegantly summarized by Maxwell's equations—is a relativistic theory. In the nonrelativistic limit, electrodynamics scales down to electrostatics; that is, in this limit not only do the effects of the finite speed of interactions vanish, but also all magnetic interactions, including magnetic induction. We will demonstrate that nonrelativistic calculations of magnetic properties are made possible by using a relativistic coupling of particles to fields. This is a perfectly valid, pragmatic approach, but implies a mixing of theories with different transformation properties and makes the underlying physics less transparent than a fully relativistic formulation.

Illustration of Anionic polythiophene acetic acid with sodium counterions in water solution.

Figure 1.6 Anionic polythiophene acetic acid with sodium counterions in water solution.

Most quantum-chemical calculations describe molecules alone in the universe and at 0 K. This may be a good approximation for dilute gases, since intermolecular interactions are often negligible unless explicitly addressed in the experimental studies. However, the scope of applications for the computational chemist is far wider, including molecules in liquid and solid phases, as well as at interfaces. With molecular response approaches, we can address cases where the properties of the macroscopic system is predominantly determined by the properties of the molecules, that is, one can regard the effects of the environment as perturbations. This is not, however, the same as saying that one employs the techniques of perturbation theory to account for these interactions; rather, self-consistent schemes are commonly adopted. Either way, the space is divided into regions treated at different levels of accuracy, where the core quantum region is large enough to encompass those parts of the electron density that give rise to the response signal in the experiment. The exterior of the core quantum region is treated by means of classical physics, either in terms of continuum dielectric medium models, such as the polarizable continuum model (PCM), or by the introduction of discrete charge multipole expansions representing the solvent molecules in molecular mechanics (MM) approaches. The latter of the two techniques can be rather straightforwardly combined with molecular dynamics (MD) simulations, be it classical or quantum MD, and it is a common procedure to extract snapshots from MD simulations and perform individual response theory calculations based on each of these snapshots. This is a way to sample the molecular configuration space under given experimental conditions of temperature, liquid, and chromophore densities, and the statistically averaged theoretical spectra represents the final result of the simulation. Figure 1.6 shows one such snapshot taken from simulations of luminescence properties of an oligothiophene chromophore used for optical probing of certain proteins. The partitioning of a solvent or a large molecular system into a target area treated accurately and with the remaining environment being treated in a more approximate manner is often referred to as focused models, and can be considered a computational equivalent of the chemical concept of active or functional regions in a complex molecular system.

This brief introduction aims to show that when the computational chemist needs to evaluate molecular response properties, a number of problems of computational and theoretical nature must be addressed, explicitly or at least assessed with respect to the expected relevance and importance. This book aims to be a valuable aid in this work, by introducing a response theory framework that is general enough to host the most commonly adopted model chemistries while leaving out most details of their implementations. The choice of basis set in the calculation is an obvious parameter that is not of much concern in the formulation of a theory, and, although less obvious, this holds largely true also for the choice of Hamiltonian. The reason for the latter simplification is that, before evaluating specific matrix elements, one can suppress the reference to the explicit form of the Hamiltonian and write it in a general form, including summations of one-electron c01-math-092 and two-electron c01-math-093 terms—a form which is common to one-, two-, and four-component Hamiltonians. Even such a seemingly insurmountable complexity as that provided by the interactions with the surrounding environment can, from our perspective, be largely ignored. The key here is to define effective operators that will couple the external classical region to the core quantum region. Once this coupling is achieved, it can be viewed as a modification of the Hamiltonian that fits into our general framework. When it comes to the third axis in Figure 1.3—the electronic structure methods used in the model chemistries—the situation becomes substantially more complicated. In this case, care has to be taken already at the outset of the derivation of the response functions when choosing an equation of motion for the time evolution of the wave function. Although all formulations of exact-state wave mechanics give rise to identical time propagations of a given initial reference state, this is no longer true in approximate-state theories. We will discuss this issue at length and consider two fundamentally different formulations of time-dependent perturbation theory. These two formulations are referred to as the Ehrenfest and the quasi-energy approaches, respectively, and while the former is applicable only to variational electronic structure methods, the latter can also be applied to nonvariational methods.

¹ Hyperpolarizability is a naming convention originally chosen on the basis of the presumed increase of the polarizability in the presence of a strong electric field, whereas hypopolarizability, a name not used any longer, was suggested for systems where there was a decrease.

Chapter 2

Quantum Mechanics

Quantum mechanics, that mysterious, confusing discipline, which none of us really understands but which we know how to use. It works perfectly, as far as we can tell, in describing physical reality, but it is a ‘counter-intuitive discipline’, as social scientists would say. Quantum mechanics is not a theory, but rather a framework, within which we believe any correct theory must fit.

M. Gell-Mann (1980)

In this chapter we give a brief, and in many respects, an incomplete introduction to quantum mechanics, with a focus on the bare minimum of concepts that are central to the subsequent parts of the book. In addition to providing the necessary basis for the subsequent chapters, this chapter introduces much of the notation that will be used throughout the book.

2.1 Fundamentals

2.1.1 Postulates of Quantum Mechanics

The theory of quantum mechanics rests on a set of postulates as illustrated in Table 2.1 in the case of a single-particle system in one-dimensional (1D) motion. The precise knowledge of the system state and physical observables in the classical case has, in the quantum case, been turned into a situation where the values of observables can only be predicted probabilistically. The coordinate vectors c02-math-001 that are often used as a basis for the infinite-dimensional Hilbert space can be imagined (but not realized) as states of perfect particle localization c02-math-002 , but in general the position of the quantum particle is described by the probability density (or simply the density)

2.3 equation

Changes in the density may be predictable and deterministic in accordance with Postulate D (the time-dependent Schrödinger equation), but they may also be indeterministic and associated with sudden quantum jumps to eigenstates of the operators c02-math-004 in accordance with Postulate C.

Table 2.1 Postulates in classical and quantum mechanics

2.1.2 Lagrangian and Hamiltonian Formalisms

From Postulate D, we see that the correspondence between classical and quantum mechanics goes through the Hamiltonian formalism. In order to better understand the construction of the Hermitian operators in quantum mechanics that are associated with physical observables, we need to take a closer look at Hamiltonian mechanics. In fact, in order to get the complete picture of quantization, we shall start from Lagrangian mechanics. A fascinating aspect of Lagrangian mechanics is that it allows us to derive the equations of motion of a dynamical system of c02-math-031 particles from a single scalar function, the Lagrangian. Let c02-math-032 collectively denote the c02-math-033 coordinates of the particles in the system—mapping out a point in the configuration space of the system—with corresponding velocities given by their total time derivatives c02-math-034 . In particular, let c02-math-035 and c02-math-036 be the particle coordinates at times c02-math-037 and c02-math-038 , respectively. We next introduce the action c02-math-039 defined as

2.4 equation

which is an example of a mathematical object known as a functional. In general, functionals map functions onto complex numbers, in contrast to functions that map sets into sets. In order to explore the possible values of the input function, a functional is often given as a definite integral over the function variables, that is

2.5

equation

The functional dependence of c02-math-042 on the function c02-math-043 is indicated by square brackets to separate it from arguments of a function that are kept inside parenthesis. In the calculus of variations, a central question is how the value of the functional depends on the variations of the function. For this purpose, the first variation of the functional c02-math-044 is defined as

2.6 equation

where c02-math-046 is an unspecified test function that can be viewed as pointing out a direction in the function space along which one takes an infinitesimal step by means of the multiplication with c02-math-047 . A variation c02-math-048 in the input function is connected to the variation of the functional through the functional derivative

2.7 equation

This can be compared to the corresponding relation for functions, where an infinitesimal change in the argument is connected to the change in the function through the (function) derivative

2.8 equation

In our applications, the function c02-math-051 is typically to be replaced by either the wave function c02-math-052 or the electron charge density c02-math-053 , and we will be scanning the respective function spaces in the search for elements that yield zero first-order variations. Since the function variations c02-math-054 are arbitrary, this implies that the functional derivative itself is zero. In the present case, the functional under study is the action c02-math-055 , which is a functional of the system trajectory c02-math-056 and expressed as an integral over the Lagrangian from an initial time c02-math-057 to a final time c02-math-058 , as seen in Eq. (2.4).

The principle of stationary action states that the actual path taken by the system between times c02-math-059 and c02-math-060 is the one for which an infinitesimal change in the trajectories leaves the action unchanged (stationary), or, in other words, the first variation of the action vanishes

2.9

equation

To find the corresponding functional derivative of c02-math-062 with respect to the trajectory c02-math-063 in configuration space, we compare the above expression with the defining relation

2.10 equation

We see that the first, but not the second term of Eq. (2.9), has the desired form. In order to bring the latter into the correct form, we carry out an integration by parts

2.11

equation

We now impose that the start- and endpoints of the trajectories are known and fixed such that

2.12 equation

Then the first term on the right-hand side of Eq. (2.11) vanishes and we obtain a form of c02-math-067 that allows us to extract the functional derivative. Setting the functional derivative to zero for all coordinates c02-math-068 gives

2.13

equation

These equations of motion are known as the Euler–Lagrange equations. The condition c02-math-070 implies that functional derivatives for individual coordinates are zero only to the extent that the coordinates are linearly independent. It may be that there are constraints on the motions of the particles; for instance, that our particles are constrained to move on the surface of a sphere, in which case this condition no longer holds. However, if there are c02-math-071 constraints on the form

2.14 equation

then we may build them into our formalism by transforming to a set of c02-math-073 linearly independent generalized coordinates c02-math-074 , collectively denoted as c02-math-075 , which now defines our configuration space. In our derivations we will for simplicity ignore such constraints.

At this point it is important to note that the Lagrangian function is not given a priori, but should be chosen to give trajectories in accordance with the experiment. Building new Lagrangians therefore means building new physics. Generally, in the absence of dissipative forces such as friction, the Lagrangian has the form c02-math-076 where c02-math-077 is the kinetic energy and c02-math-078 is called the generalized potential. The kinetic energy is the Lagrangian of free particles, corresponding to the case c02-math-079 . In the absence of any constraints, the kinetic energy is a function of velocities only and we may re-write the Euler–Lagrange equations, Eq. (2.13), in the form of Newton's second law as

2.15 equation

where the generalized force c02-math-081 appears

2.16 equation

Now let us consider the force c02-math-083 acting on a single particle. The work done by the force upon the particle along its trajectory between times c02-math-084 and c02-math-085 is given by a line integral

2.17 equation

A first thing to note is that if we use Newton's second law c02-math-087 , we can show that the total work along the trajectory is equal to the change in kinetic energy c02-math-088 between the endpoints of the trajectory, that is,

2.18

equation

Let us assume that the force is monogenic, that is, derivable from a single potential function c02-math-090 , as in Eq. (2.16). We can then write the infinitesimal work as

2.19

equationequation

where, in the final step, we have used the total differential

2.20

equation

of the generalized potential. In many situations, but not in the presence of electromagnetic forces, the potential function has no explicit dependence on velocities and time. In such a case, the work along the trajectory is given by the value of the potential function at the endpoints, that is,

2.21 equation

The work is therefore path independent and will for instance be zero around any closed path. Combining the above result with Eq. (2.19) we have

2.22 equation

and the force is said to be conservative.

Returning to the general case, let us consider the total time derivative of the work. Starting from Eq. (2.19), we obtain

2.23

equation

Keeping in mind that c02-math-096 , we can rearrange this to

2.24 equation

We identify the expression in square brackets with energy

2.25

equation

where c02-math-099 is the potential energy.¹ Eq. (2.24) therefore demonstrates that the total energy is conserved if the general potential has no explicit time dependence.

Let us now turn to Hamiltonian mechanics. Again, a scalar function, this time the Hamiltonian, takes center stage. It can be obtained from the Lagrangian by a Legendre transformation

2.26 equation

A crucial difference between the Hamiltonian and the Lagrangian is that the dependence on velocity c02-math-101 in the latter is replaced by dependence on momentum c02-math-102 in the former, achieved through the Legendre transformation. We can also use the Legendre transformation to find the general definition of momentum: The Hamiltonian does not depend on velocity and so by taking the partial derivative with respect to velocity c02-math-103 on both sides of Eq. (2.26), we obtain

2.27

equation

If the generalized potential does not depend on velocity, that is, for a conservative system, the momentum reduces to its usual (nonrelativistic) form c02-math-105 , denoted by the mechanical momentum. However, as discussed in Chapter 3, magnetic forces depend on particle velocities, and hence the need to start from the correct Lagrangian to get a proper definition of momentum for use in the Hamiltonian, both in classical and quantum mechanics.

Hamilton's equations, which appear in Postulate D in Eq. (2.1), can be obtained from the Legendre transformation, Eq. (2.26), as well. Taking the partial derivative with respect to momentum we obtain the first equation. Taking the partial derivative with respect to coordinate c02-math-106 and using the Euler–Lagrange equations, Eq. (2.13), gives the second equation. Finally, taking the total time derivative of the Hamiltonian using the chain rule, we obtain

2.28 equation

By using Hamilton's equations, we find that the first two terms cancel. The final result is

2.29 equation

where the final step is obtained by taking the explicit time derivative of the Legendre transformation. If the Lagrangian has no explicit time dependence, the above expression shows that the Hamiltonian is time independent and thus a constant of motion. In fact, we can obtain Eq. (2.29) directly by taking the total time derivative of the Lagrangian. For a single particle, we get

2.30

equation

where the final step is obtained using the Euler–Lagrange equation, Eq. (2.13). By rearrangement, we obtain

2.31 equation

We see that the Legendre transformation leading to the Hamiltonian function drops out directly in the above expression. We can even go one step further and show that it is equivalent to the energy expression of Eq. (2.25) by writing the Legendre transformation, Eq. (2.26), as

2.32 equation

and carrying it out separately for the components c02-math-112 and c02-math-113 of the Lagrangian. From the kinetic energy, we get

2.33 equation

and from the generalized potential we get directly the potential energy. The Hamiltonian can therefore be identified with the total energy,² which is conserved if there is no explicit time dependence in the generalized potential.

Let us illustrate these concepts with a very simple, albeit not very chemical, example, namely that of the free particle. We can deduce the nonrelativistic (NR) form of this Lagrangian from some simple considerations: (i) for a free particle there is no preferred moment in time (homogeneity of time), so the Lagrangian cannot depend explicitly on time, and (ii) there is no preferred point or direction in space (homogeneity and isotropy of space) so the Lagrangian cannot depend explicitly on the position or the velocity direction. Based upon these considerations, we write the Lagrangian as proportional to the square of the velocity (or speed) of the particle

2.34 equation

where the constant of proportionality is fixed by experiment. Using Eq. (2.27), we find that the momentum of the free particle is c02-math-117 . Next, we obtain the corresponding free-particle Hamiltonian by the Legendre transformation, Eq. (2.26),

2.35 equation

Note that the intermediate expression, referred to as the energy function, does not represent a valid Hamiltonian since it is expressed in terms of the velocity. Only after the velocity is substituted by the momentum does the Hamiltonian become a function of position and momentum as required. The velocity in terms of momentum is provided by the first of Hamilton's equations, whereas the second takes the form c02-math-119 , corresponding to Newton's first law. We again stress the fact that the momentum is defined from the Lagrangian and will therefore hinge upon a correct definition of this function. This point will turn out to be crucial when considering the introduction of electromagnetic fields, as will be discussed in Chapter 3.

The final quantum-mechanical Hamiltonian is obtained by replacing the position and momentum variables by their corresponding operators. These should be chosen such that the canonical commutation relation

2.36 equation

is obeyed. In the coordinate representation, we use the substitutions

2.37 equation

Another option is the momentum representation

2.38 equation

but it is less useful for localized systems such as molecules.

The relativistic (R) free-particle case is more complicated and the appropriate form of the Lagrangian is less evident. In fact, since there is no longer a universal time, we should not single out a specific infinitesimal of time c02-math-123 in the action [Eq. (2.4)] but rather use the Lorentz-invariant proper time c02-math-124 , which is the time in the reference frame moving with the particle. The variation of the action becomes

2.39

equation

where the relativistic Lagrangian c02-math-126 is now a function of the four-position c02-math-127 , four-velocity c02-math-128 , and the proper time c02-math-129 . Following the same line of arguments as for the nonrelativistic free particle, it can be proposed that the relativistic Lagrangian should be proportional to the square of the four-velocity. In fact, the suitable form is c02-math-130 , which clearly is Lorentz invariant. We can go to a specific frame by noting that

2.40 equation

where the Lorentz factor c02-math-132 was defined in Eq. (1.11) and can be considered a diagnostic for relativistic effects. Relativistic effects will be important (and c02-math-133 large) when the speed of the particle c02-math-134 is sizable compared to the speed of light, here denoted by c02-math-135 .

The action integral can now be re-expressed in its familiar form of Eq. (2.4) with the relativistic free-particle Lagrangian

2.41 equation

At first sight, Eqs. (2.34) and (2.41) look very different, but if we expand the inverse Lorentz factor to second order in c02-math-137 , we obtain

2.42 equation

The first term, which does not contribute to the equation of motion since it is constant, is minus the rest mass, the second term is the nonrelativistic free-particle Lagrangian c02-math-139 , Eq. (2.34), and the third term represents a relativistic correction known as the mass-velocity correction. Using Eq. (2.27), we obtain the relativistic momentum

2.43 equation

which differs from the nonrelativistic definition by the Lorentz factor. The appearance of the Lorentz factor can be associated with a relativistic mass increase. A Legendre transformation of the Lagrangian in accordance with Eq. (2.26) gives

2.44 equation

Before the identification of a valid Hamiltonian can be made, we must in this expression replace the velocity hidden in the Lorentz factor by the momentum. However, in order to carry out this replacement, we must proceed via a quadratic form and arrive at

2.45

equation

This is a problematic expression, for many reasons. One is that we face the possibility of free particles with negative sign. Another is that quantization is not straightforward because the momentum operator appears in the square root. We shall, however, postpone this discussion to Section 3.2.2.

2.1.3 Wave Functions and Operators

The wave function c02-math-143 associated with a single electron is, from a mathematical point of view, an element of a Hilbert space, which is a complete vector space equipped with a norm in the form of a scalar product. The probabilistic interpretation of the wave function implies that it should belong to the Hilbert space c02-math-144 of square-integrable functions. More precisely, it can be enforced that

2.46

equation

Starting from the vector space c02-math-146 associated with the scalar orbital part, we may successively add further degrees of freedom through a sequence of direct products

2.47 equation

The electron spin is typically added by multiplying the spatial orbital by spin functions c02-math-148 and c02-math-149 , but we may also proceed in a more formal manner by using their representations

2.48 equation

The corresponding representation of the spin operator c02-math-151 is given by c02-math-152 , where the Pauli spin matrices appear

2.49

equation

If we go all the way to the description of the electron provided by the Dirac equation, we also have to include a further two-dimensional part c02-math-154 , now associated with the charge conjugation degrees of freedom (changing the sign of the charge but leaving other properties intact) because the Dirac equation describes both electrons and their antiparticles, the positrons. The four components of the Dirac wave function is constructed as

2.50

equationequation

forming the general wave function as the sum of these four components to arrive at

2.51 equation

The label c02-math-157 of each component c02-math-158 refers to spin. It is important to realize, though, that the label c02-math-159 , referring to large and small components, respectively, does not correspond directly to electronic or positronic degrees of freedom. The entire wave function is in fact either electronic or positronic, but related to that of its antiparticle through charge conjugation. Arriving at the full four-component form of the wave function, we note that the adopted ordering of vector spaces in Eq. (2.47), with c02-math-160 to the left of c02-math-161 , leads to a collection of the two large (small) components in the upper (lower) half of the spinor.

In a two-component formalism, one suppresses the degrees of freedom associated with c02-math-162 , and, in a one-component nonrelativistic formalism, one may also suppress the reference to c02-math-163 and instead introduce and manage electron spin in a more ad hoc manner, as described above. Because the nonrelativistic Hamiltonian is spin independent, one can always separate the spatial and spin parts of the wave function, but regardless of such simplifications in the representation and notation of the wave function, the true nature of the electronic wave function is described by Eq. (2.51), and we refer to such a wave function as a four-spinor.

The corresponding density is

2.52

equation

We will, however, often use the simplified notation of the density as introduced in Eq. (2.3) regardless of the number of components in the wave function.

We extract information that can be related to experiment from wave functions by forming integrals

2.53 equation

In particular, expectation values ( c02-math-166 ) connect to observables, whereas the squared norm of transition moments ( c02-math-167 ) connects to spectral intensities. The right-hand side of Eq. (2.53) stresses that the operator c02-math-168 acts to the right, on the ket. The adjoint operator c02-math-169 is defined to give the same result when acting on the bra, that is

2.54

equation

In more informal usage one writes

2.55 equation

that is, an operator becomes its adjoint when acting to the left.³ Quantum-mechanical operators are selected to be self-adjoint ( c02-math-174 ) such that expectation values are real. We usually refer to such operators as Hermitian, and we shall adhere to this usage, although there is a subtle difference between the two terms.⁴

Similar to what we discussed for wave functions, general electronic quantum-mechanical operators are written as a direct product between one part that acts in orbital space c02-math-182 , one that acts in spin space c02-math-183 , and one part associated with the charge conjugation degrees of freedom

2.56 equation

The formation of a product operator means that we take the product in each individual operator space in accordance with

2.57 equation

A small selection of operators is given in Table 2.2. In this table, only one of the Cartesian components is provided for the vectorial operators, but other components follow suit.

Table 2.2 A selection of quantum mechanical operators

Operators relating to Cartesian components c02-math-221 and c02-math-222 are obtained by straightforward generalization.

With pure spatial (or spin) operators, we refer to situations when only c02-math-223 (or c02-math-224 ) is different from the identity operator—the identity operator itself is both a pure spatial and a pure spin operator. It is common practice to allow for simplified notations of operators, for example, to omit the symbol for the direct product or, for pure operators, to omit the identity operators. We shall use such conventions whenever there is no risk of confusion. An example of a pure orbital operator where the identity operators in c02-math-225 and c02-math-226 have been left out is given by the nonrelativistic Hamiltonian in the form given in Eq. (2.35). In dealing with pure operators, simplifications may be introduced in the notation, and it also makes it possible to separately treat symmetries in the vector and operator spaces. In the nonrelativistic domain, we do so for instance by the construction of spin-adapted wave functions and operators as well as by the use of symmetry-adapted spatial orbitals. In the relativistic domain, on the other hand, spin and spatial degrees of freedom are always coupled with the spin-orbit interactions and, as a consequence, it becomes impossible to separately treat spin and spatial symmetries.

The generalization of the vector space and the operators to a many-electron system is quite straightforward. For instance, the state vectors for two-electron systems are elements of a Hilbert space of the form

2.58 equation

and the corresponding operators are given by

2.59 equation

For many-electron systems, we should keep in mind that electrons belong to the family of fermions and as such they obey the Pauli exclusion principle. This means that no two interacting electrons can be in the same quantum state, and the vector space c02-math-229 is therefore limited to include only antisymmetric two-particle vectors of the type

2.60

equation

In the last step we have made use of a determinant to write the antisymmetrized two-particle state, and we have also included an optional, explicit, normalization constant of c02-math-231 together with the assumption that the one-particle state vectors are individually normalized. Such a determinant is known as a Slater determinant in quantum chemistry, and one of the most notable properties of the Slater determinant is that when c02-math-232 , the wave function vanishes. The Slater determinant thus ensures that the Pauli principle is satisfied. It is clear that the set of all possible Slater determinants constructed from a complete set of one-particle states constitutes the natural basis of c02-math-233 .

Another aspect to note—and which is a reminder in the handling of the vector as well as the operator spaces—is that there are vectors and operators that themselves may not be written as direct products. The general electronic wave function in Eq. (2.51) is one example as it is a sum of four direct product components, and for many-electron systems, a simple example is given by the coupled two-particle operator with a coordinate representation in terms of the Dirac delta function

2.61 equation

Let us assume that c02-math-235 forms a complete set of orthonormal one-particle functions. We then have

2.62

equation

which provides a resolution of c02-math-237 into products of one-particle functions. Another and also very important example is given by an instantaneous Coulomb interaction operator. The Coulomb operator for a two-electron system takes the basic form

2.63 equation

We may replace the orthogonality condition for the basis vectors with a condition of biorthogonality

2.64 equation

or, equivalently,

2.65 equation

where the operator c02-math-241 relates the pairs of biorthogonal vectors

2.66 equation

We can then proceed to obtain a resolution of the Coulomb operator according to

2.67

equationequation

which provides a resolution of the Coulomb operator in a one-particle basis. Apart from providing a demonstration of how to decouple a two-particle interaction operator, it gives a glimpse into a family of techniques developed to achieve cost-efficient numerical treatments of two-electron integrals.

In this section, we have demonstrated how to construct wave functions and operators for many-electron systems. We emphasize that the foundations of quantum mechanics do not depend on the inclusion or exclusion of relativity, and to the extent possible, the aim has been to give a presentation that is valid in both situations. At the same time, it cannot be avoided that, at a certain level of detail, one must be specific about the choice of Hamiltonian (and other operators) and depending on that choice, there are particular and less general aspects to be explored and utilized in the calculations.

2.2 Time Evolution of Wave Functions

The time evolution of the state vector is given by the time-dependent Schrödinger equation, Postulate D, and is thus dictated by the Hamiltonian. Formally, we may describe the evolution of the wave function from some initial time c02-math-244 by the introduction of a time evolution operator, or time propagator, according to

2.68 equation

An equation of motion for the propagator is obtained by first inserting the above expression into the time-dependent Schrödinger equation. Since c02-math-246 is independent of the time variable c02-math-247 , we obtain a resulting operator equation

2.69 equation

which is equivalent to the time-dependent Schrödinger equation. Expressions for the time evolution operator are now obtained by integrating this equation. Alternatively, we can proceed via an infinitesimal time evolution operator. For an infinitesimal time interval c02-math-249 , we may Taylor expand the wave function to first order and use the time-dependent Schrödinger equation to arrive at

2.70

equation

which shows that the infinitesimal time evolution operator is

2.71 equation

Since we only retain the infinitesimal interval c02-math-252 to first order and since the Hamiltonian is Hermitian, it follows that the infinitesimal time evolution operator is unitary

2.72

equation

For a finite time interval, we may express the time evolution operator as a product of infinitesimal time evolution operators by dividing the time interval into sufficiently small pieces

2.73 equation

which yields a time propagator that reads as

2.74

equation

This product form of individually unitary operators shows that the time evolution for a finite interval is unitary as well. When the Hamiltonian is time independent, the time evolution operator attains a simple exponential form

2.75

equation

If the initial state happens to be an eigenstate of the Hamiltonian, the time dependence becomes an overall phase factor of the wave function. Let us assume that at time c02-math-257 , the wave function is an eigenstate of c02-math-258 denoted by c02-math-259 with eigenvalue c02-math-260 . From Eq. (2.68), we get

2.76 equation

and the particle density, Eq. (2.3), is thus in this case time independent

2.77 equation

For this reason, the eigenstates of the Hamiltonian are also referred to as the stationary states of the system and are characterized by being separable in space and time. It is important to note that even for a system in a stationary state there can be an associated nonzero linear momentum as illustrated by the case of the nonrelativistic free particle. The stationary states can in this case be written as

2.78 equation

and the associated linear momentum is given by

2.79 equation

This corresponds to particle motion at a speed of c02-math-265 for the stationary state. For a molecular system with bound electrons, a nonzero linear momentum may be found in orbitally degenerate states—such as the c02-math-266 -states in hydrogen, or the c02-math-267 -states in benzene—but the current density is such that the particle flow into a probing infinitesimal volume c02-math-268 equals the flow out of